Inverse spectral problem for the density of a vibrating elastic membrane

This paper is concerned with the recovery of an unknown symmetric density function in the weighted Helmholtz equation with Dirichlet boundary conditions from the lowest few eigenvalues. By using the piecewise constant function to approximate the density function and using the Rayleigh-Ritz approach to discretize the differential equation, the continuous inverse eigenvalue problem is converted to a related matrix inverse eigenvalue problem and then a least squares problem for the discrete model is formulated. The solution of the least squares problem via an iterative method is discussed and then an approximation to the unknown density is recovered. Numerical experiments are given to confirm its competitiveness.